Let's say we have a conventional two-bladed fixed-pitch propeller. All things being equal, would the same propeller but with half the size -- 50% radius, have 50% of the performance of the first propeller?
Would the propeller hub in the middle be the same size? You did say all things being equal, which I read as "unchanged".
Example, 86" total diameter prop, 12" diameter hub, leaving 2 blades that are (86 -12)/2 = 37" from edge of hub to tip of blade, and 74" of working blade length.
Then you reduce the radius to 43" with same 12" hub, so you have (43 - 12)/2 = 15.5" blade from hub to tip, and 31" of working blade length. 31" is less then 1/2 of 74", so you've lost more than 50%.
Assuming the engine can't spin any faster due to the engine governor limiting the RPMs, you're not getting off the ground.
Short answer. No. In engine test cells, where torque measurement is not a criteria, techs utilize what is called a club propeller. Club props are usually large 4 blade props which would be seen on engines larger than the one being tested. The difference being the club prop has had a significant length of each blade trimmed away. Why? The club prop is very good at absorbing power while generating proportionally much less thrust (Low Efficiency). Hopefully this at least partially addresses your question.